BEAM IMPEDANCE. Olav Berrig / CERN. Lanzhou China May 2018 CERN-ACC-SLIDES /05/2018

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1 BEAM IMPEDANCE Olav Berrig / CERN CERN-ACC-SLIDES /05/2018 Lanzhou China May

2 Outline 1. What is beam impedance? 2. Beam impedance is modelled as a lumped impedance 3. New formula for longitudinal beam impedance 4. Panofsky-Wenzel theorem and transverse impedance 5. Lab measurements of beam impedance 2

3 What is beam impedance? Beam impedance is just a normal impedance. However, it is very difficult to understand beam impedance because it is not a lumped impedance but measured over a length. In addition it is defined as the difference in impedance between an accelerator equipment and a straight vacuum chamber. The straight vacuum chamber must have constant cross-section; have the same length as the accelerator equipment and have walls that are superconducting (also called perfectly conducting PEC). A particle moving in a straight vacuum chamber with constant crosssection and superconducting walls have no beam 3 impedance.

4 What is beam impedance? An accelerator without beam impedance does not have instabilities. Beam impedance is not our friend! Beam impedance gives the beam a kick i.e. a disturbing force acting on the beam. The beam impedance forces will make the beam oscillate, just like a mass suspended between springs: NB! Landau damping is not shown because it is not damping, in spite of the name! Synchrotron radiation Beam Damping kicker 4

5 What is beam impedance? An example of transverse impedance, that gives the beam a transverse kick! Here measured with the beam Andrea Latina Hao Zha s Ref [1]

6 What is beam impedance? There are many types of beam impedance: Beam impedance from the currents in the walls of accelerator equipment (beam coupling impedance): 1) Resistive wall impedance 2) Geometric impedance Space charge beam impedance 1) Direct space charge impedance 2) Indirect space charge impedance Damping kicker impedance, Electron cloud, impedance, 6

7 What is beam impedance? There are many types of beam impedance: Beam impedance from the currents in the walls of accelerator equipment (beam coupling impedance): 1) Resistive wall impedance 2) Geometric impedance Space charge beam impedance 1) Direct space charge impedance 2) Indirect space charge impedance In the following, I will only talk about beam coupling impedance Damping kicker impedance, Electron cloud, impedance, 7

8 What is beam impedance? The wall currents must oppose the beam current, so that the fields outside the vacuum chamber are zero 8

9 What is beam impedance? Equipment PEC vacuum pipe When we calculate the beam impedance for an equipment, we compare the equipment to a perfectly conducting (PEC) vacuum chamber with the same dimensions at start and end. 9

10 What is beam impedance? Classical thick wall regime: R = w L Current density estimation J [A/m 2 ] skin depth This area represents the difference between superconducting (PEC) vacuum chamber and one with resistance. b b+ B [Tesla] m 0 I 2pb b b+ Curtesy of M.Migliorati 10

11 What is beam impedance? Skin depth: ( 60, 30) (-60, 30) (-60,-30) ( 60,-30) (-40, 7) ( 40, 7) (-40, -7) ( 40, -7) Ohm CST_freq Theory HFSS_freq CST_Wakefield Collimator: Length: 200 mm Width: 120 mm Height: 60 mm Electrical conductivity of jaws: σ = 100 S/m Rad s 11

12 Beam impedance: R + j w L versus R i w L Circuit definition American Fourier Chinese and European Fourier R + j w L R + j w L R i w L In my experience, accelerator components have only resistive and inductive coupling impedance. 12

13 Beam impedance modelled as lumped impedance Definition of beam impedance: = Voltage over equipment Drive particle act as a current. (It s a Dirac delta function) Definition of lumped impedance: Z(w) Dirac Delta, where h(t) h(t) = impulse response 13

14 Beam What impedance is beam impedance? modelled by lumped impedance The wake function W (t) is the equipment response function, i.e. the response to a Dirac delta function. The impedance is, according to normal theory, just the Fourier transform of the response function: 14

15 Beam What impedance is beam impedance? modelled by lumped impedance In other texts (See e.g. Ref. [6]) one will often find this definition: 15

16 Beam impedance modelled by lumped impedance Wall currents generate electro-magnetic fields i.e. photons when bend along the cavity walls. Photons The electro-magnetic fields stays in the cavity and generates a resonance, which will disturb i.e. kick the following bunch. A resonance is modeled as a RLC-circuit: RLC-circuit definition used for resonance ( American Fourier) Z (w) = ò 0 k loss = 1 2p W (t )e jwt dt ò 0 Â{ Z (w)}dw w 0 = 1 LC ; Q = R C L R Z (w) = 1+ jq(w w 0 -w 0 w) k loss = w 0 4 R Q The bigger R/Q the bigger the energy loss. NB! This definition of the loss factor is only valid for a bunch that is a dirac delta function. The more general definition will be given later. The energy lost, is equal to the loss factor k loss multiplied with the square of the charge of the bunch: 16

17 New formula for longitudinal beam impedance The Longitudinal beam impedance is a function of the transverse position of the drive and test particles i.e. 4 variables. It can therefore be decomposed into 15 parameters (Z0, Z1 xd, Z1 xt, etc..) that represent all combinations of the 4 variables: 17

18 New formula for longitudinal beam impedance Holomorphic decomposition: Any two dimensional field, and very importantly a field that can really exist (so not an artificially constructed field), can be decomposed into multipolar components. This is the same idea used in Fourier transforms. The holomorphic decomposition expands the field into normal and skew multipolar functions: Zero order first order second order NB! Notice that the coefficients for x squared and y squared are same numerical value but opposite signs 18

19 New formula for longitudinal beam impedance The normal and skew multipolar functions are well known from accelerator magnets: Zero order first order second order 19

20 New formula for longitudinal beam impedance Using the holomorphic decomposition for both the drive and test particles, knowing that the coefficients for the squared values of xd & yd and xt & yt must be of opposite sign, the formula can be reduced to 13 terms: 20

21 New formula for longitudinal beam impedance Using a property, called the Lorentz reciprocity principle, which says that if we exchange the positions of the drive and test particles, the beam impedance stays unchanged, i.e.. This leads to 5 equalities: The new formula for longitudinal beam impedance finally has only 8 terms: 21

22 New formula for longitudinal beam impedance Using a property, called the Lorentz reciprocity principle, which says that if we exchange the positions of the drive and test particles, the beam impedance stays unchanged, i.e.. This leads to 5 equalities: The new formula for longitudinal beam impedance finally has only 8 terms: Quadrupolar term 22

23 New formula for longitudinal beam impedance Using a property, called the Lorentz reciprocity principle, which says that if we exchange the positions of the drive and test particles, the beam impedance stays unchanged, i.e.. This leads to 5 equalities: The new formula for longitudinal beam impedance finally has only 8 terms: Quadrupolar term Dipolar terms H & V 23

24 What is beam impedance? The longitudinal beam impedance have 8 parameters Interchanging the drive and test particles, will give the same beam impedance. It is caused by the Lorentz reciprocity theorem (well known to RF people as the identity S21 S12): 24

25 What is beam impedance? The longitudinal beam impedance have 8 parameters The Lorentz reciprocity theorem is responsible for coupling primary and secondary windings in a transformer: 25

26 What is beam impedance? The longitudinal beam impedance have 8 parameters The Lorentz reciprocity theorem is responsible for coupling primary and secondary windings in a transformer: This is why the name of a beam impedance that is generated by the wall currents is a beam coupling impedance 26

27 What is beam impedance? The longitudinal beam impedance have 8 parameters The new formula shows that 90 degree symmetrical structures only have dipolar impedance and that this impedance is the same in all directions 27

28 New formula for longitudinal beam impedance This new formula is not valid for resonances nor for non-relativistic beams, because both are spread out in 3D. The formula is practically valid for beams with, even though theoretically there will always be other terms, but these terms are proportional to, so will not be important in practice: 28

29 Panofsky-Wenzel and transverse impedance The rigid bunch approximation states that the beam motion is little affected during the passage through the structure. So the beam shape is rigid and it always moves unchanged with the bunch. Wakefield The force acting on the test particle: Using Maxwell s equations: 29

30 Panofsky-Wenzel and transverse impedance The force acting on the test particle: Very important: Because the wakefield is only a function of s then: B(s) This leads to Position of drive particle: Position of the test particle: When inserting the partial differentials on the right, the terms in the bracket cancels out and gives zero. 30

31 Panofsky-Wenzel and transverse impedance The force acting on the test particle: Very important: Because the wakefield is only a function of s then: B(s) This leads to Position of drive particle: Position of the test particle: Panofsky Wenzel theorem When inserting the partial differentials on the right, the terms in the bracket cancels out and gives zero. 31

32 Panofsky-Wenzel and transverse impedance To obtain the theorem in terms of impedance, one can simply start from the wake function form: Then change the derivative with the time derivative. Use : Finally take the Fourier transform on both sides: NB! The transverse impedance is defined with a complex factor: Panofsky Wenzel theorem 32

33 Panofsky-Wenzel and transverse impedance Using the following definitions: and 33

34 Panofsky-Wenzel and transverse impedance Using the following definitions: and Panofsky Wenzel theorem In differential form 34

35 Panofsky-Wenzel and transverse impedance Because of the rigid bunch approximation, which states that the beam motion is little affected during the passage through a structure, the wake field is the same before and after the passage of an equipment. Therefore, it is as if B is only a function of s. A criterion for the Panofsky-Wenzel theorem is therefore that the vacuum chamber has to have the same cross-section before and after the equipment otherwise the B-field is not the same. 35

36 Lab measurements of beam impedance. Wire #1 We can measure the beam impedance with wire measurements This is based on the assumption that a bunch interacts with an equipment in exactly the same way as a coaxial cable (i.e. a wire inside the equipment): Ultra-relativistic beam field TEM mode coax waveguide See A.Mostacci: 36

37 Lab measurements of beam impedance. Wire #2 Network analyzer Port 1 dz/dl Network analyzer Port 2 DUT=Device under test wire dz/dl wire REF = Reference = PEC vacuum 37 chamber same length as DUT

38 Lab measurements of beam impedance. Wire #3 Beam impedance: This is the improved log formula, which is used for wire measurements 38

39 Lab measurements of beam impedance. Wire #4 39

40 Lab measurements of beam impedance. Wire #5 B 50 Ω B 50 Ω 180 o Hybrid A B C D A B C D 40

41 Lab measurements of beam impedance. Wire #6 Characteristic impedance distance between them Two wire measurement give only the dipolar impedance of two wires, each with diameter and with is (See Example: = 10.0 mm Z = 120/1. ln(40) ~ 450 Ohm d = 0.5 mm i.e. 225 Ohm per wire Subtract 50 Ohm, as usual, this gives 175 Ohm per wire. So it is always 175 Ohm per wire 41 independent of the chamber diameter!

42 Lab measurements of beam impedance. Wire #7 Two wire measurement give only the dipolar impedance +a -a = dipolar impedance The distance between the wires is 2 a: A B C D A B C D 42

43 Another What is measure beam impedance? of transverse beam impedance! An example of transverse impedance, that gives the beam a transverse kick! Here measured with the beam Andrea Latina Hao Zha s Ref [1]

44 Lab measurements of beam impedance. Wire #8 Easy method to firmly straighten the wire. Make hole in connector and solder a thin wire to the resistor. This method was invented by Muzhaffar Hazman 44

Make hole in connector and solder a thin wire to the resistor.

45 Lab measurements of beam impedance. Wire #8 Easy method to firmly straighten the wire. Make hole in connector and solder a thin wire to the resistor. When soldering the resistor to the connector, keep the other soldering cold, otherwise it will dissolve. Use plier as heat sink. This method was invented by Muzhaffar Hazman 45

46 Lab What measurements is beam impedance? of beam impedance. Wire #9 46

47 Lab measurements of beam impedance. Wire #10 ab-note pdf T. Kroyer, F. Caspers, E. Gaxiola MKE Kicker measurements 47

48 Lab measurements of beam impedance. Wire #11 An example of serigraphy in the SPS Extraction Kicker Magnets (SPS-MKE) 48

49 Lab measurements of beam impedance. Wire #12 Kicker Transition piece, i.e. keep electrical connection with the vacuum chamber 49

50 Lab measurements of beam impedance. Wire #13 Kicker Transition piece, i.e. keep electrical connection with the vacuum chamber 50

Lab measurements of beam impedance. Wire #14 Collimator measurement https://indico.cern.

51 Lab measurements of beam impedance. Wire #14 Collimator measurement /attachments/ / /SLAC_RC_SPS_pla n.pdf N. Biancacci, P. Gradassi, T. Markiewicz, S. Redaelli, B. Salvant, G. Valentino 51

52 Lab measurements of beam impedance. Probe #1 52

53 Lab measurements of beam impedance. Probe #2 53

54 Lab measurements of beam impedance. Probe #3 54

55 Lab measurements of beam impedance. Probe #4 smith chart 55

56 Lab measurements. Measure Q reflection. Probe #5 A resonance is a circle in the smith diagram. Three different types of Q: 1) The loaded Q (QL) 2) The unloaded Q (Q0) 3) The Q of the external world (Qext ). We want Q 0, but we can only measure Q L and b: 56

57 Lab measurements. Measure Q reflection. Probe #6 Ql β 2 Q Ql β 2 Q Courtesy of C.Vollinger and T.Kaltenbacher 57

58 Beam impedance presentation. Lanzhou - China 感谢您的关注 58

59 [1] Measurement of transverse kick in CLIC accelerating structure in FACET Hao Zha, Andrea Latina, Alexej Grudiev [2] Holomorphic decomposition. John Jowett \\cern.ch\dfs\projects\ilhc\mathematicaexamples\accelerator\multipolefields.nb [3] On single wire technique for transverse coupling impedance measurement. H. Tsutsui [4] Longitudinal instability of a coasting beam above transition, due to the action of lumped discontinuities V. Vaccaro [5] Wake Fields and Instabilities Mauro Migliorati [6] G.Rumolo, CAS Advanced Accelerator Physics Trondheim, Norway August [7] THE STRETCHED WIRE METHOD: A COMPARATIVE ANALYSIS PERFORMED BY MEANS OF THE MODE MATCHING TECHNIQUE M.R.Masullo, V.G.Vaccaro, M.Panniello [7] Two Wire Wakefield Measurements of the DARHT Accelerator Cell. Scott D. Nelson, Michael Vella [8] Shunt impedance, RLC-circuit definition, Accelerator definition, Alexej Grudiev [9] A.Mostacci [10] COUPLING IMPEDANCE MEASUREMENTS: AN IMPROVED WIRE METHOD V.Vaccaro [11] Interpretation of coupling impedance bench measurements H. Hahn [12] Measurement of coupling impedance of accelerator devices with the wire-method J.G. Wang, S.Y. Zhang 59

60 [13] Longitudinal and Transverse Wire Measurements for the Evaluation of Impedance Reduction Measures on the MKE Extraction Kickers. Kroyer, T ; Caspers, Friedhelm ; Gaxiola, E 60

61 Energy loss when beam pass through an equipment Shunt impedance, RLC-circuit definition, Accelerator definition, Alexej Grudiev In the following, only beam coupling impedance are calculated. Beam coupling impedance is generated by the currents in the walls of the equipment, and is the only significant impedance in high energy accelerators. 61

62 Beam impedance modelled by lumped impedance Shunt impedance, RLC-circuit definition, Accelerator definition, Alexej Grudiev 62

63 What is beam impedance? The longitudinal beam impedance have 8 parameters The beam impedance is now decomposed into 13 parameters : Z [xd, xt, yd, yt] = Z0 +Z1XD xd+z1xt xt+z1yd yd+z1yt yt +Z2XYDXYD (xd 2 -yd 2 )+Z2XYTXYT (xt 2 - yt 2 ) +Z2XDXT xd xt+z2xdyd xd yd+z2xdyt xd yt +Z2XTYD xt yd+z2xtyt xt yt+z2ydyt yd yt The new formula is identical to the previous from Tsutsui: 63

64 What is beam impedance? The longitudinal beam impedance have 8 parameters xd=0,yd=0.0020,xt=0.0025,yt=0 xd=0.001,yd=0.0025,xt=0,yt=0 xd=0.0025,yd=0,xt=0,yt= xd=0,yd=0,xt=0.001,yt= Real Imaginary Real Imaginary Interchanging the drive and test particles always give the same beam impedance. CST 64

65 What is beam impedance? The longitudinal beam impedance have 8 parameters Z [xd, xt, yd, yt] = Z0 +Z1X (xd+xt)+z1y (yd+yt) +Z2XYDTXYDT (xd 2 +xt 2 -yd 2 - yt 2 ) +Z2XDTYDT (xd yd+xt yt) +Z2XDTYTD (xd yt +xt yd) +Z2XDXT xd xt+z2ydyt yd yt 65

66 What is beam impedance? CST Wakefield example illustrating the 8 parameters Z2YTIm,Z2YDIm Z2YTRe,Z2YDRe Prediction: Z2XYDTXYDT (xd 2 +xt 2 -yd 2 - yt 2 ) CST Z2XTRe,Z2XDRe Z2XTIm,Z2XDIm 66

67 What is beam impedance? CST Wakefield example illustrating the 8 parameters Prediction: Z2XDTYDT (xd yd+xt yt) CST Z2XTYTIm, Z2XDYDIm Z2XTYTRe, Z2XDYDRe 67

68 What is beam impedance? CST Wakefield example illustrating the 8 parameters Prediction: Z2XDTYTD (xd yt +xt yd) CST Z2XTYDRe, Z2XDYTRe,Z2XTYDIm, Z2XDYTIm 68

69 What is beam impedance? CST Wakefield example illustrating the 8 parameters Prediction versus simulation 3 examples Example 1: yd=2.5, yt=2.5 Example 2: xd=1.0, yd=2.5 Example 3: xd=-1.5, yd=2.0, xt=2.5 CST Prediction Real Prediction Imaginary Simulation Real Simulation Imaginary 69

70 What is beam impedance? The rotating wire method One wire represents the drive particle and the other wire represents the test particle. In this measurement, we do not have a positive current in one wire and a negative current in the other Both wires are measured individually i.e. single-ended In the additional slide, it is demonstrated how this measurement can derive all 8 parameters

71 What is beam impedance? The rotating wire method Some implications of the new 8 parameter formula: 1) Transverse impedance The offset term is not automatically zero, depends on the shape of the equipment 2) Transverse impedance Is it possible to shape a collimator e.g. in three-fold symmetric form so that its transvers impedance is zero up to second order? 3) Transverse impedance The beam oscillates during instability, is it possible to shape equipment 71 in such a way that the drive position works against the instability?

72 Supporting material for slide 72

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CERN Accelerator School Wakefields. Prof. Dr. Ursula van Rienen, Franziska Reimann University of Rostock

CERN Accelerator School Wakefields Prof. Dr. Ursula van Rienen, Franziska Reimann University of Rostock Contents The Term Wakefield and Some First Examples Basic Concept of Wakefields Basic Definitions